The invention relates generally to processing of numerical data which characterize subsurface earth formations. More particularly, the invention relates to a method and a system for optimizing the injection and production well fluid allocation factors used in a material balance analysis of a hydrocarbon reservoir.
In the commercial recovery of hydrocarbons it is desirable to estimate the fluid saturations and pressure changes that occur in the reservoir as a result of injecting fluids into the reservoir and producing fluids therefrom, and then compare these results with actual measurements to maximize the efficiency of recovery. A key constraint in determining accurate estimates is the conservation of total mass of injected and produced fluids, i.e., the "material balance."
As used in the art, the term "material balance" describes the process of determining the total fluid volumes entering and leaving a volume over a time period using this information to compute resulting reservoir pressures and fluid saturations. Material balance calculations are well known and described fully in Petroleum Reservoir Engineering: Physical Properties, J. W. Amyx, D. M. Bass, Jr., R. L. Whiting, McGraw-Hill Book Co., New York, 1960, pp. 561-598, and Applied Petroleum Reservoir Engineering, B. C. Craft and M. F. Hawkins, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1959, pp 148-156 (hereinafter collectively referred to as "Material Balance References"). Generally, the reservoir is divided into a set of volumes called "patterns" centered on producing wells with injection wells on the pattern borders. A separate material balance calculation is performed for each pattern. Because injection wells are on pattern boundaries, fluid from a single injection well must be allocated to more than one pattern. Current practice is to estimate the allocation of fluids from each of the injection wells surrounding a producer by assigning allocation factors for each well, where the allocation factors represent the fraction of fluid injected to or produced from a well, into or out of a well pattern. For production wells, the production allocation factors describe the split of produced fluids among production zones vertically (e.g., two production zones would result in two unknown allocation factors for each production well). Injection well allocation factors describe the portion of injected fluid that migrates to each of the surrounding production wells for each production zone. For example, an injection well completed in two production zones and bordering on four patterns will result in eight unknown allocation factors.
Mathematical solution of the fluid allocation problem is complex because, for a typical reservoir, there can be hundreds to thousands of unknown allocation factors. Reservoir pressure response to injection and production is non-linear making the allocation factors difficult to estimate by traditional optimization techniques. When comparing estimates to actual data, changing a well allocation in one pattern to match field observations changes the well allocation factors in all surrounding patterns. Traditional practice is to manually iterate possible solutions for the allocation of fluids from each of the injection wells surrounding a producer until a "reasonable" pressure profile for all patterns in the reservoir is achieved. This is a labor intensive and subjective process. Efforts to automate this allocation process using a least-squares, linear programming approach have not been satisfactory.
By way of further background, optimization methods known as "genetic algorithms" have been applied to non-linear problems in many diverse areas, including operation of a gas pipeline, factory scheduling and semiconductor layout. Genetic algorithms serve to select a string (referred to as a "chromosome") of numbers ("genes") having values ("alleles") that provides the optimum value of a "fitness function." According to this technique, a group of chromosomes (a "generation") is first randomly generated, and the fitness function is evaluated for each chromosome. A probability function is then produced to assign a probability value to each of the chromosomes according to its fitness function value, so that a chromosome with a higher fitness function value obtains a higher probability. A "reproduction pool" of chromosomes is then produced by random selection according to the probability function; the members of this reproduction pool are more likely to be selected from the higher fitness function values. A randomly selected chromosome from the reproduction pool then "reproduces" with another, randomly selected, chromosome in the reproduction pool by exchange of genes at a randomly selected "crossover" point in the chromosome. This reproduction is repeated to generate a second generation of chromosomes. Mutation may be introduced by randomly altering a small fraction of the genes in the second generation (e.g., one in one thousand). The fitness function is then evaluated for each of the chromosomes in the second generation, and the reproduction process is repeated until the desired convergence is obtained.
Over the years, researchers have developed many different variants of the original Genetic Algorithm implementation. The algorithm described above is one implementation that was used to solve the material balance problem. A second variant of the Genetic Algorithm methodology, a "Bit Climber," was also applied. The Bit Climber provided a solution with accuracy equal to that achieved with the first Genetic Algorithm procedure and improved computer speed. Other variants of the Genetic Algorithm methodology can also be applied to this problem. Techniques for implementing the Bit Climber are well known to those skilled in the art and are described in Lawrence Davis, "Bit-Climbing, Representational Bias and Test Suite Design", Proceedings of the Fourth International Conference on Genetic Algorithms, University of California, San Diego, Jul. 13-16, 1991, p. 18-23, which is hereby incorporated by reference in its entirety.
What is needed, therefore, is a method of automating the material balance process as it relates to the estimate of fluid allocation factors for production and injection wells in the reservoir, using a stochastic optimization technique, such as a genetic algorithm procedure, that can more readily accommodate non-linear aspects of this problem.